Limit Cycles Coming from Some Uniform Isochronous Centers

This article is about the weak 16th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centers (x) over dot = -y + x(2)y(x(2) + y(2))(n), (y) over dot = x + xy(2) (x(2) + y(2))n, of degree 2n + 3 and we perturb them inside the class of all polynomial differential systems of degree 2n + 3. For n = 0, 1 we provide the maximum number of limit cycles, 3 and 8 respectively, that can bifurcate from the periodic orbits of these centers using averaging theory of first order, or equivalently Abelian integrals. For n = 2 we show that at least 12 limit cycles can bifurcate from the periodic orbits of the center.